Cartan Matrices and Dynkin Diagrams

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 27 July 2012

Cartan matrices and Dynkin diagrams

The Cartan matrix is $$C=\left(⟨{\alpha }_i^{\vee },{\alpha }_i⟩\right)\text{with}{\alpha }_i^{\vee }=\frac{2{\alpha }_i}{⟨{\alpha }_i,{\alpha }_i⟩}$$ So that the matrix of the form is $$A=\left(⟨{\alpha }_i,{\alpha }_j⟩\right)=DC\text{where}D=diag\left(\frac{⟨{\alpha }_i,{\alpha }_i⟩}{2}\right).$$

Type $A_{n-1}$: The Dynkin diagram is

and $$C=\left(\begin{array}{cccc}\begin{array}{ccc}2& -1& \\ -1& 2& -1\\ & -1& 2& -1\\ & & & \ddots \end{array}& & 0& \\ 0& & \begin{array}{cc}2& -1\\ -1& 2\end{array}\end{array}\right)=A.$$

Type $A_5$: The Dynkin diagram is

and

$$Q=\left(\begin{array}{ccccc}0& v-u& 1& 1& 1\\ u-v& 0& v-u& 1& 1\\ 1& u-v& 0& v-u& 1\\ 1& 1& u-v& 0& v-u\\ 1& 1& 1& u-v& 0\end{array}\right)$$

Type $B_n$: The Dynkin diagram is

and $$C=\left(\begin{array}{cccc}\begin{array}{ccc}2& -2& \\ -1& 2& -1\\ & -1& 2& -1\\ & & & \ddots \end{array}& & 0& \\ 0& & \begin{array}{cc}2& -1\\ -1& 2\end{array}\end{array}\right)\text{and}A=\left(\begin{array}{cccc}\begin{array}{ccc}2& -2& \\ -2& 4& -2\\ & -2& 4& -2\\ & & & \ddots \end{array}& & 0& \\ 0& & \begin{array}{cc}4& -2\\ -2& 4\end{array}\end{array}\right),$$ and $D=diag\left(1,2,2,\dots ,2\right)$.

Type $B_5$: The Dynkin diagram is

and

$$Q=\left(\begin{array}{ccccc}0& v-u^2& 1& 1& 1\\ u-v^2& 0& v-u& 1& 1\\ 1& u-v& 0& v-u& 1\\ 1& 1& u-v& 0& v-u\\ 1& 1& 1& u-v& 0\end{array}\right)$$

Type $C_n$: The Dynkin diagram is

and $$C=\left(\begin{array}{cccc}\begin{array}{ccc}2& -1& \\ -2& 2& -1\\ & -1& 2& -1\\ & & & \ddots \end{array}& & 0& \\ 0& & \begin{array}{cc}2& -1\\ -1& 2\end{array}\end{array}\right)\text{and}A=\left(\begin{array}{cccc}\begin{array}{ccc}4& -2& \\ -2& 2& -1\\ & -1& 2& -1\\ & & & \ddots \end{array}& & 0& \\ 0& & \begin{array}{cc}2& -1\\ -1& 2\end{array}\end{array}\right),$$ and $D=diag\left(2,1,1,\dots ,1\right)$.

Type $C_5$: The Dynkin diagram is

and

$$Q=\left(\begin{array}{ccccc}0& v^2-u& 1& 1& 1\\ u^2-v& 0& v-u& 1& 1\\ 1& u-v& 0& v-u& 1\\ 1& 1& u-v& 0& v-u\\ 1& 1& 1& u-v& 0\end{array}\right)$$

Type $D_n$: The Dynkin diagram is

and $$C=\left(\begin{array}{cccc}\begin{array}{ccc}2& 0& -1\\ 0& 2& -1\\ -1& -1& 2& -1\\ & & -1& 2& -1\\ & & & & \ddots \end{array}& & 0& \\ 0& & \begin{array}{cc}2& -1\\ -1& 2\end{array}\end{array}\right)=A.$$

Type $D_5$: The Dynkin diagram is

and

$$Q=\left(\begin{array}{ccccc}0& 1& v-u& 1& 1\\ 1& 0& v-u& 1& 1\\ u-v& u-v& 0& v-u& 1\\ 1& 1& u-v& 0& v-u\\ 1& 1& 1& u-v& 0\end{array}\right)$$

Type $E_6$: The Dynkin diagram is

and $$C=\left(\begin{array}{cccccc}2& 0& -1& 0& 0& 0\\ 0& 2& 0& -1& 0& 0\\ -1& 0& 2& -1& 0& 0\\ 0& -1& -1& 2& -1& 0\\ 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& -1& 2\end{array}\right)=A\text{and}Q=\left(\begin{array}{cccccc}0& 1& v-u& 1& 1& 1\\ 1& 0& 1& v-u& 1& 1\\ u-v& 1& 0& v-u& 1& 1\\ 1& u-v& u-v& 0& v-u& 1\\ 1& 1& 1& u-v& 0& v-u\\ 1& 1& 1& 1& u-v& 0\end{array}\right).$$

Type $E_7$: The Dynkin diagram is

and $$C=\left(\begin{array}{ccccccc}2& 0& -1& 0& 0& 0& 0\\ 0& 2& 0& -1& 0& 0& 0\\ -1& 0& 2& -1& 0& 0& 0\\ 0& -1& -1& 2& -1& 0& 0\\ 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& 0& -1& 2\end{array}\right)=A\text{and}Q=\left(\begin{array}{ccccccc}0& 1& v-u& 1& 1& 1& 1\\ 1& 0& 1& v-u& 1& 1& 1\\ u-v& 1& 0& v-u& 1& 1& 1\\ 1& u-v& u-v& 0& v-u& 1& 1\\ 1& 1& 1& u-v& 0& v-u& 1\\ 1& 1& 1& 1& u-v& 0& v-u\\ 1& 1& 1& 1& 1& u-v& 0\end{array}\right).$$

Type $E_8$:

and $$C=\left(\begin{array}{cccccccc}2& 0& -1& 0& 0& 0& 0& 0\\ 0& 2& 0& -1& 0& 0& 0& 0\\ -1& 0& 2& -1& 0& 0& 0& 0\\ 0& -1& -1& 2& -1& 0& 0& 0\\ 0& 0& 0& -1& 2& -1& 0& 0\\ 0& 0& 0& 0& -1& 2& -1& 0\\ 0& 0& 0& 0& 0& -1& 2& -1\\ 0& 0& 0& 0& 0& 0& -1& 2\end{array}\right)=A\text{and}Q=\left(\begin{array}{cccccccc}0& 1& v-u& 1& 1& 1& 1& 1\\ 1& 0& 1& v-u& 1& 1& 1& 1\\ u-v& 1& 0& v-u& 1& 1& 1& 1\\ 1& u-v& u-v& 0& v-u& 1& 1& 1\\ 1& 1& 1& u-v& 0& v-u& 1& 1\\ 1& 1& 1& 1& u-v& 0& v-u& 1\\ 1& 1& 1& 1& 1& u-v& 0& v-u\\ 1& 1& 1& 1& 1& 1& u-v& 0\end{array}\right).$$

Type $F_4$: The Dynkin diagram is

and $$C=\left(\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -2& 0\\ 0& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right)\text{and}A=\left(\begin{array}{cccc}2& -1& 0& 0\\ -1& 2& -2& 0\\ 0& -2& 4& -2\\ 0& 0& -2& 4\end{array}\right)\text{and}Q=\left(\begin{array}{cccc}0& v-u& 1& 1\\ u-v& 0& v-u^2& 1\\ 1& u-v^2& 0& v-u\\ 1& 1& u-v& 0\end{array}\right)$$ and $D=diag\left(1,1,2,2\right)$.

Type $G_2$: The Dynkin diagram is

and $$C=\left(\begin{array}{cc}2& -3\\ -1& 2\end{array}\right),A=\left(\begin{array}{cc}2& -3\\ -3& 6\end{array}\right),Q=\left(\begin{array}{cc}0& v-u^3\\ u-v^3& 0\end{array}\right),\text{with}\left(\begin{array}{cc}1& 0\\ 0& 3\end{array}\right).$$ and $D=diag\left(1,1,2,2\right)$.

Type $A_5^{\left(1\right)}$:The Dynkin diagram is

and

$$Q=\left(\begin{array}{ccccc}0& v-u& 1& 1& u-v\\ u-v& 0& v-u& 1& 1\\ 1& u-v& 0& v-u& 1\\ 1& 1& u-v& 0& v-u\\ v-u& 1& 1& u-v& 0\end{array}\right)$$

Passing to the nonsimply laced case (following Reinecke and Lusztig's book), $$\tilde{C}={\left({\tilde{a}}_{ij}\right)}_{i,j\in \tilde{I}}\text{an indecomposable Cartan matrix.}$$ $\tilde{C}$ is a quotient of a simply laced Cartan matrix $C={\left(a_{ij}\right)}_{i,j\in I}$ under the action of a cyclic group $Z$ on $I$. Then $$\tilde{I}=\left\{\text{orbits of} Z \text{on} I\right\}.$$ There is an induced action of $Z$ on $U^{+}$ and on $B\left(\infty \right).$ Then $$\tilde{B}=\left\{Z \text{fixed points in} B\left(\infty \right)\right\}$$ $$\begin{array}{rcl}{\tilde{f}}_i\tilde{b}={\tilde{b}}^{\prime }& \iff & \text{there is a sequence} b^{\prime }={\tilde{f}}_{i_r}\cdots {\tilde{f}}_{i_1}b \text{in} B\left(\infty \right)\\ & & \text{such that} \left\{i_1,\dots ,i_r\right\} \text{is a} Z-\text{orbit in} I.\end{array}$$ In this form $C_n$ comes from $A_{2n-1}$:

$$\stackrel{\mathbb{Z}/2\mathbb{Z}=Z}{\to }$$

$B_n$ comes from $D_{n+1}$:

$$\stackrel{\mathbb{Z}/2\mathbb{Z}=Z}{\to }$$

$F_4$ comes from $E_6$:

$$\stackrel{\mathbb{Z}/2\mathbb{Z}=Z}{\to }$$

$G_2$ comes from $D_4$:

$$\stackrel{\mathbb{Z}/3\mathbb{Z}=Z}{\to }$$

$A_{l-1}^{\left(1\right)}$ comes from $A_{\infty }$:

$$\stackrel{\mathbb{Z}\mathbb{Z}=Z}{\to }$$

References

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

[DHL] H.-D. Doebner, Hennig, J. D. and W. Lücke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.

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